In parallel, we also measure the velocity autocorrelation function (VACF), \(C_v(t)=\langle \vec{v}(t)\cdot \vec{v}(0)\rangle \)  that quantifies the temporal velocity correlation over a time interval \(t\) . For Brownian particles, VACF satisfies an exponential decay [1]}, [2]}, [3]}, \(C_v(t)=e^{-t/\tau }\) , where \(\tau =m/\zeta \)  represents the typical time scale [1]}. Figures 2d-f show VACF measured for all datasets. At short times, the VACF follows exponential decay as Brownian particles. However, VACF becomes negative for a longer time, implying attractive social interactions among individuals. We find the empirical VACF can be fit by \(Ae^{-t/\tau }-Be^{-t/\tau ^{\prime }}\) , where former and later terms capture the thermal diffusion and social interaction, respectively. Note that \(\tau ^{\prime }\)  diffs from \(\tau _s\)  measured in MSD but they are positively correlated (see SM Section B1).
